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The Multipole Model and Refinement

Jyväskylä Summer School on Charge Density August 2007. The Multipole Model and Refinement. Louis J Farrugia. Jyväskylä Summer School on Charge Density August 2007. Spherical Atom Scattering.

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The Multipole Model and Refinement

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  1. Jyväskylä Summer School on Charge Density August 2007 The Multipole Model and Refinement Louis J Farrugia

  2. Jyväskylä Summer School on Charge Density August 2007 Spherical Atom Scattering The “standard” crystallographic refinement programs use a model of atomic scattering based on spherical atoms. The scattering from these atoms is isotropic. The example below is for Chromium (Z = 24) Static atom Uiso = 0.03Å2

  3. Jyväskylä Summer School on Charge Density August 2007 The Pseudo-atom Multipole Representation Spherical core Spherical valence Deformation valence The most commonly used formalism for describing aspherical atomic densities (and hence scattering) is the Hansen-Coppens pseudo-atom model. The total crystal density is modelled by the sum of pseudo-atoms at the atomic sites. Core - (potentially) refinable population Spherical valence - refinable monopole population Pv (charge) and kappa  Deformation valence – comprises a radial part and a spherical-harmonic part- refinable multipole populations Plm and kappa ' N.K. Hansen & P. Coppens (1978), Acta Cryst. A34, 909.

  4. Jyväskylä Summer School on Charge Density August 2007 Real Spherical Harmonics dipoles quadrupoles octupoles hexadecapoles Z axis is vertical, green is +ve, red is -ve

  5. Jyväskylä Summer School on Charge Density August 2007 Real Spherical Harmonics Spherical harmonics used in multipole models are density normalised for l = 0, i = 1; for l > 0, i = 2 This normalisation means that for a spherically symmetric function, a population parameter of 1.0 denotes an electron population of 1.0 For the non-spherical functions, with l > 0, which have both positive and negative lobes, the population parameter represents the number of electrons shifted from the negative to the positive regions In the special case of sites with cubic symmetry, the spherical-harmonic basis functions become mixed, and so-called Kubic Harmonics are then required. P. Coppens (1997), “X-ray Charge Densities and Chemical Bonding”, IUCr Monograph, OUP, Oxford

  6. Jyväskylä Summer School on Charge Density August 2007 Choice of the Radial Functions The choice of the radial basis is in principle arbitrary, except that the analytical angular behaviour requires Rnlmr-1to be finite at the origin. In practice either Gaussian or Slater type functions have been used. The XD program uses (as one option - CSZD) these Slater-type functions : Default values of the al and n(l) parameters for each atomic type are stored in databanks. Derived from atomic wavefunction calculations. May be changed by user intervention. K. Kurki-Suonio (1977) Isr. J. Chemistry16, 132.

  7. Jyväskylä Summer School on Charge Density August 2007 Choice of the Radial Functions in XD In the XD program, these radial functions are specified by the user in the MASTER FILE XD.MAS !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! <<< X D MASTER FILE >>> $Revision: 4.07 (Apr 25 2003)$ 03-05-03 ! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! TITLE H2C2O4 CELL 6.1024 3.4973 11.9586 90.0000 105.7710 90.0000 WAVE 0.7107 LATT C P SYMM 0.5000 - X, 0.50000 + Y, 0.50000 - Z BANK CR !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! MODULE *XDLSM SELECT model 4 2 1 1 based_on F^2 test SELECT cycle 10 dampk 1. cmin 0.6 cmax 1. eigcut 1.d-09 SAVE deriv lsqmat cormat SOLVE *inv diag *cond !------------------------------------------------------------------------------ SCAT CORE SPHV DEFV 1S 2S 3S 4S 2P 3P .... 6D 5F DELF' DELF'' NSCTL O CHFW CHFW CSZD 2 -2 0 0 -4 0 .... 0 0 0.0106 0.0060 0.580 C CHFW CHFW CSZD 2 -2 0 0 -2 0 .... 0 0 0.0033 0.0016 0.665 H CHFW CHFW CSZD -1 0 0 0 0 0 .... 0 0 0.0000 0.0000 -0.374 END SCAT The type of databank The types of radial function may be individually selected for the core, spherical valence and deformation valence respectively

  8. Jyväskylä Summer School on Charge Density August 2007 Radial Functions for the Core Density CHFW Electron density from full HF expansion limited to the core electrons defined in the SCAT table fcore RDTB Also possible to construct core scattering from a table (seriously limits possibilities in analysis) K RDTB CHFW CSZD 2 2 2 -1 6 6 .... 0 0.1410 0.1580 0.371 18.00000 17.64523 16.66577 15.27737 13.73497 12.24942 10.94846 9.88004 9.03673 8.38210 7.87046 7.45841 7.11031 6.79966 6.50846 6.22570 5.94559 5.66597 5.38698 5.11011 4.83746 4.57129 4.31370 4.06651 3.83114 3.60864 3.39964 3.20446 3.02313 2.85543 2.70095 2.55916 2.42940 2.31096 2.20308 2.10498 2.01590 1.93508 1.86179 1.79534

  9. Jyväskylä Summer School on Charge Density August 2007 Radial Functions for Spherical Valence Density CHFW Electron density from full HF expansion limited to the valence electrons defined in the SCAT table fvalence SCAT CORE SPHV DEFV 1S 2S 3S 4S 2P C chfw chfw cszd 2 -2 0 0 -2 (2j0(2s2s) + 2j02p2p))/4

  10. Jyväskylä Summer School on Charge Density August 2007 Radial Functions for Deformation Density CSZD A single- z Slater function will be used. The z exponent is constructed from the best single- z of the valence orbitals. Radial node-less function of an atomic orbital Radial node-less density function of an atomic orbital The nl values must satisfy Poisson’s equation. The conditions are nl l E. Clementi and D. L. Raimondi (1963)J. Chem. Phys. 38, 2686.

  11. Jyväskylä Summer School on Charge Density August 2007 Radial Functions for Deformation Density CHFW Electron density from full HF expansion C CHFW CHFW CHFW 2 -2 0 0 -2 .... 0 CHFW (2s2s)+(2p2p) 1 CHFW (2s2s) 2 CHFW (2p2p) 3 RDSD 3 4.4 4 CSZD This defines the second monopole to be identical to the SPHV Use the density of 2s orbital Use the density of 2p orbital Use a single-, but with modified nl and a Must define the form for all l values Use default single- (2j0(2s2s) + 2j02p2p))/4

  12. Jyväskylä Summer School on Charge Density August 2007 Radial Functions for Deformation Density 3d orbital of Fe CHFW - full expansion (5 Slater functions) less sensitive to deformations, more adequate in describing a molecular orbital which closely resembles an atomic orbital (“low overlap regime”) CSZD - single Slater function more expanded, therefore more sensitive to deformations, less adequate to describe a “low overlap regime” Tanaka et al.(1986)J. Chem. Phys.12, 6969.

  13. Jyväskylä Summer School on Charge Density August 2007 Radial Functions for Transition Metals 3d radial extension (relatively contracted) 4s radial extension (highly diffuse) distance from nucleus Only few reflections (often affected by extinction & absorption errors) contain information on 4s electrons. scattering curve for density from 4s orbital sin(q)/l

  14. Jyväskylä Summer School on Charge Density August 2007 Radial Functions for Transition Metals SCAT CORE SPHV DEFV 1S 2S 3S 4S 2P 3P 4P 3D Fe CHFW CHFW CHFW 2 2 2 2 6 6 0 -6 0 CHWF (3d3d) 1 RDSD 4 2.0 2 CHWF (3d3d) 3 CHWF 4 2.0 4 CHWF (3d3d) 4s in “core” d orbitals in SPHV Entry for Fe atom in parameter file XD.INP Fe 3 2 2 12 5 0 3 3 4 1 0 0.489010 0.453599 0.063590 1.0000 0.005450 0.005063 0.004324 -0.000270 0.000747 0.000968 6.00000.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 (starting) value for Pv P00 not refined

  15. Jyväskylä Summer School on Charge Density August 2007 Radial Functions for Transition Metals SCAT CORE SPHV DEFV 1S 2S 3S 4S 2P 3P 4P 3D Fe CHFW CHFW CHFW 2 2 2 0 6 6 0 -8 0 CHWF (4s4s) 1 RDSD 4 2.0 2 CHWF (3d3d) 3 CHWF 4 2.0 4 CHWF (3d3d) remove 4s from “core” Try refining 4s occupation…. only d orbitals in SPHV even order multipoles are produced by d orbitals odd order multipoles are produced by s-d mixing (should be small anyway) and therefore are more diffuse Entry for Fe atom in parameter file XD.INP Fe 3 2 2 12 5 0 3 3 4 1 0 0.489010 0.453599 0.063590 1.0000 0.005450 0.005063 0.004324 -0.000270 0.000747 0.000968 8.00000.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 (starting) value for Pv (starting) value for P00

  16. Jyväskylä Summer School on Charge Density August 2007 Physical Importance of the nl Parameters For elements of the third period (Si, S, P, Cl), an improved agreement is often found when the radial exponents for higher multipoles are larger than those expected based on atomic orbitals. Increasing the value of nlmoves the maximum further from the nucleus. The higher multipoles model the density in the interatomic regions – the covalent electron density nl =4 nl = 6 nl = 8

  17. Jyväskylä Summer School on Charge Density August 2007 Physical Importance of the  parameters The kappa parameters are scaling parameters for the al values. They are very important for obtaining a good fit (take into account differing effective atomic charges), but their refinement is difficult.  =1.2  = 1.0  = 0.8

  18. Jyväskylä Summer School on Charge Density August 2007 The Kappa Restricted Multipole Model The problems experienced with the refinement of the kappa parameters has led to the concept of the Kappa Restricted Multipole Model (KRMM). In this model, the kappa parameters for the the deformation valence () are derived from mutipole refinements using theoretical (error-free) structure factors obtained from high quality wavefunctions. These (and more) parameters are now incorporated into databases. A. Volkov, Y. A. Abramov & P. Coppens (2001)Acta Cryst. A57, 272. P. M. Dominiak, A. Volkov, X. Li, M. Messerschmidt, P. Coppens (2007) J. Chem. Theory Comp.3, 232

  19. Jyväskylä Summer School on Charge Density August 2007 Choice of the Databank The traditional choice is the databank derived from the Clementi-Roetti table. These were based on Roothan-Hartree Fock calculations on ground state isolated atoms and relevant ions. Each atomic orbital is expanded in a series of Slater functions The use of analytical expressions to compute the scattering factors and the density means that all properties may be computed analytically as well f Total core valence sin()/ E. Clementi & C. Roetti, (1974). At. Data Nucl. Data Tables, 14, 177.

  20. Jyväskylä Summer School on Charge Density August 2007 Choice of the Databank For the heavier elements (Z > 36, Kr)the effects of relativistic contractions cannot be neglected, especially for core electrons. For these elements, it is prefereable to use a wave function that mimics the atomic relativistic density. In XD this is the SCM database (H to Xe), or the VM database (H to Cf) Difference is not large, but will have an effect on the refined thermal parameters Relativistic Non-relativistic 1s electron density of Xe difference distance from nucleus (Å) Su, Z.; Coppens, P. Acta Cryst1997,A53, 749, Su, Z.; Coppens, P. Acta Cryst1998,A54, 646, Macchi, P.; Coppens, P. Acta Cryst., 2001, A57, 656

  21. Jyväskylä Summer School on Charge Density August 2007 Choice of the Databank Discrepancies between the relativistic and non-relativistic scattering factors increase with the resolution of the data. These scattering factors should be used for elements in the 5th period (2nd row transition metals) . The main advantages are 1. more accurate thermal parameters 2. better treatment of the core density P. Macchi & P. Coppens (2001) Acta Cryst., A57, 656 P. Macchi et al (2001) J. Phys. Chem. A.105, 9231

  22. Choice of the Databank xd.bnk_RHF_CR: (BANK CR) CHFW Non relativistic wave functions (H-Kr, including ions) Clementi, E. & Roetti, C. (1974). At. Data Nucl. Data Tables, 14, 177-478 RDSD E. Clementi and D. L. Raimondi, J. Chem. Phys. 38, 2686-2689 (1963). Analytical Fit : International Tables for Crystallography xd.bnk_RHF_BBB: (BANK BBB) CHFW Non relativistic wave functions (H-Xe) C. F. Bunge, J. A. Barrientos, A. V. Bunge At. Data Nucl. Data Tables, 53, 113-162 (1993). RDSD E. Clementi and D. L. Raimondi, J. Chem. Phys. 38, 2686-2689 (1963). Analytical Fit : International Tables for Crystallography xd.bnk_RDF_SCM: (BANK SCM) CHFW Relativistic wave functions (H-Xe, including ions) Z. Su and P. Coppens Acta Cryst., A54, 646 (1998): P. Macchi and P. Coppens Acta Cryst., A57, 656 (2001). RDSD E. Clementi and D. L. Raimondi, J. Chem. Phys. 38, 2686-2689 (1963). Analytical Fit : Su, Z.; Coppens, P. Acta Cryst1997,A53, 749, Macchi, P.; Coppens, P. Acta Cryst., 2001, A57, 656 xd.bnk_PBE-QZ4P-ZORA: (BANK VM) CHFW Relativistic wave functions (H-Cf) unpublished RDSD E. Clementi and D. L. Raimondi, J. Chem. Phys. 38, 2686-2689 (1963). Analytical Fit : Macchi, P.; Volkov, A. unpublished

  23. Jyväskylä Summer School on Charge Density August 2007 The Refinable Atomic Parameters SHELX x,y,z, occupancy, Uiso (or U11 U22 U33 U12 U13 U23) - maximum 10 parameters/atom XD x,y,z, Uiso (or U11 U22 U33 U12 U13 U23) (9 parameters/atom) Anharmonic Gram-Charlier coefficients 3rd + 4th order CjklDjklm (25 parameters/atom) PvP00 P10 P11± P20 P21± P22± P30 P31± P32± P33± P40 P41± P42± P43± P44± 1 1 3 5 7 9 = 26 multipoles maximum 60 parameters/atom Neither possible nor desirable to refine 60 parameters/atom ! 1. even with high resolution, usually results in a too low data/parameter ratio 2. least-squares refinement will not be stable – too strong correlations between parameters – e.g. between anharmonic thermal parameters and multipole populations Solution: Start with a restricted model, and gradually increase the complexity.

  24. Jyväskylä Summer School on Charge Density August 2007 Refinement strategy using XDLSM Start from a refined model based on a spherical atom refinement (SHELX/CRYSTALS etc) 1. Refine scale factor KEEP KAPPA 1 2 3 KEEP CHARGE GROUP1 WEIGHT -2.0000 0.0000 0.0000 0.0000 0.0000 0.3333 SKIP OBSMIN 0. *SIGCUT 3. SNLMIN 0. SNLMAX 2. DMSDA 1.0 1.8 FOUR FMOD1 4 2 0 0 FMOD2 -1 2 0 0 KEY xyz --U2-- ----U3---- ------U4------- M- -D- --Q-- ---O--- ----H---- O(1) 000 000000 0000000000 000000000000000 00 000 00000 0000000 000000000 N(1) 000 000000 0000000000 000000000000000 00 000 00000 0000000 000000000 C(1) 000 000000 0000000000 000000000000000 00 000 00000 0000000 000000000 H(1) 000 000000 0000000000 000000000000000 00 000 00000 0000000 000000000 H(2) 000 000000 0000000000 000000000000000 00 000 00000 0000000 000000000 H(3) 000 000000 0000000000 000000000000000 00 000 00000 0000000 000000000 KAPPA 000000 KAPPA 000000 KAPPA 000000 KAPPA 000000 EXTCN 0000000 OVTHP 0 SCALE 1 END KEY

  25. Jyväskylä Summer School on Charge Density August 2007 Refinement strategy using XDLSM Start from a refined model based on a spherical atom refinement (SHELX/CRYSTALS etc) 1. Refine scale factor 2. Refine scale factor and positional parameters (non-H atoms) KEEP KAPPA 1 2 3 KEEP CHARGE GROUP1 WEIGHT -2.0000 0.0000 0.0000 0.0000 0.0000 0.3333 SKIP OBSMIN 0. *SIGCUT 3. SNLMIN 0. SNLMAX 2. DMSDA 1.0 1.8 FOUR FMOD1 4 2 0 0 FMOD2 -1 2 0 0 KEY xyz --U2-- ----U3---- ------U4------- M- -D- --Q-- ---O--- ----H---- O(1) 111 000000 0000000000 000000000000000 00 000 00000 0000000 000000000 N(1) 111 000000 0000000000 000000000000000 00 000 00000 0000000 000000000 C(1) 111 000000 0000000000 000000000000000 00 000 00000 0000000 000000000 H(1) 000 000000 0000000000 000000000000000 00 000 00000 0000000 000000000 H(2) 000 000000 0000000000 000000000000000 00 000 00000 0000000 000000000 H(3) 000 000000 0000000000 000000000000000 00 000 00000 0000000 000000000 KAPPA 000000 KAPPA 000000 KAPPA 000000 KAPPA 000000 EXTCN 0000000 OVTHP 0 SCALE 1 END KEY

  26. Jyväskylä Summer School on Charge Density August 2007 Refinement strategy using XDLSM Start from a refined model based on a spherical atom refinement (SHELX/CRYSTALS etc) 1. Refine scale factor 2. Refine scale factor and positional parameters (non-H atoms) 3. Refine scale factor, positional parameters & thermal parameters (non-H atoms) KEEP KAPPA 1 2 3 KEEP CHARGE GROUP1 WEIGHT -2.0000 0.0000 0.0000 0.0000 0.0000 0.3333 SKIP OBSMIN 0. *SIGCUT 3. SNLMIN 0. SNLMAX 2. DMSDA 1.0 1.8 FOUR FMOD1 4 2 0 0 FMOD2 -1 2 0 0 KEY xyz --U2-- ----U3---- ------U4------- M- -D- --Q-- ---O--- ----H---- O(1) 111 111111 0000000000 000000000000000 00 000 00000 0000000 000000000 N(1) 111 111111 0000000000 000000000000000 00 000 00000 0000000 000000000 C(1) 111 111111 0000000000 000000000000000 00 000 00000 0000000 000000000 H(1) 000 000000 0000000000 000000000000000 00 000 00000 0000000 000000000 H(2) 000 000000 0000000000 000000000000000 00 000 00000 0000000 000000000 H(3) 000 000000 0000000000 000000000000000 00 000 00000 0000000 000000000 KAPPA 000000 KAPPA 000000 KAPPA 000000 KAPPA 000000 EXTCN 0000000 OVTHP 0 SCALE 1 END KEY

  27. Jyväskylä Summer School on Charge Density August 2007 Treatment of Hydrogen Atoms The H atom positional parameters obtained from a spherical refinement will be incorrect (a) If neutron diffraction data are available, use the positional parameters for H atoms (b) Otherwise use the RESET BOND instruction to set X-H distances to standard neutron determined values, and refine an isotropic thermal parameter KEEP KAPPA 1 2 3 KEEP CHARGE GROUP1 WEIGHT -2.0000 0.0000 0.0000 0.0000 0.0000 0.3333 SKIP OBSMIN 0. *SIGCUT 3. SNLMIN 0. SNLMAX 2. DMSDA 1.0 1.8 FOUR FMOD1 4 2 0 0 FMOD2 -1 2 0 0 RESET BOND N(1) H(1) 1.0 etc KEY xyz --U2-- ----U3---- ------U4------- M- -D- --Q-- ---O--- ----H---- O(1) 111 111111 0000000000 000000000000000 00 000 00000 0000000 000000000 N(1) 111 111111 0000000000 000000000000000 00 000 00000 0000000 000000000 C(1) 111 111111 0000000000 000000000000000 00 000 00000 0000000 000000000 H(1) 000 100000 0000000000 000000000000000 00 000 00000 0000000 000000000 H(2) 000 100000 0000000000 000000000000000 00 000 00000 0000000 000000000 H(3) 000 100000 0000000000 000000000000000 00 000 00000 0000000 000000000 KAPPA 000000 KAPPA 000000 KAPPA 000000 KAPPA 000000 EXTCN 0000000 OVTHP 0 SCALE 1 END KEY

  28. Jyväskylä Summer School on Charge Density August 2007 Treatment of Hydrogen Atoms The H atom isotropic thermal parameters are only poor approximations (a) If neutron diffraction data are available, use the anisotropic thermal parameters (b) They will need to be scaled to the adp’s of non-H atoms (using the UIJXN program) KEY xyz --U2-- ----U3---- ------U4------- M- -D- --Q-- ---O--- ----H---- O(1) 111 111111 0000000000 000000000000000 00 000 00000 0000000 000000000 N(1) 111 111111 0000000000 000000000000000 00 000 00000 0000000 000000000 C(1) 111 111111 0000000000 000000000000000 00 000 00000 0000000 000000000 H(1) 000 000000 0000000000 000000000000000 00 000 00000 0000000 000000000 . . END KEY If using neutron parameters, they must be fixed, i.e. not refined Replace these with the exact neutron parameters XD.INP parameter file H(1) 1 2 2 4 3 1 4 4 1 1 0 0.211760 0.258310 0.138580 1.0000 0.040036 0.000000 0.000000 0.000000 0.000000 0.000000 0.7956 0.0000 0.1828 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 Replace these with the scaled neutron parameters R. H. Blessing (1995). Acta Cryst, B51, 816.

  29. Jyväskylä Summer School on Charge Density August 2007 Treatment of Hydrogen Atoms The reason that the treatment of H atoms is very important is that the charge density parameters and thermal parameters are strongly correlated. It is impossible to obtain accurate mutipole parameters without a reasonable estimate of the H atom thermal motion (Hirshfeld). H atoms have large amplitude anisotropic motion. Only required data is a CIF file with the anisotropic thermal parameters of all the non-H atoms. Method is only valid if there is no internal motion, i.e. need a rigid molecule. F. Hirshfeld (1976) Acta Cryst, 32, 239 A. Ø. Madsen (2006) J. Appl. Cryst.39, 757. – SHADE server http://shade.ki.ku.dk/ A. A. Whitten & M. A. Spackman (2006) Acta Cryst.B62, 875. –Uses ONIOM calculation – most rigourous

  30. Jyväskylä Summer School on Charge Density August 2007 Refinement strategy for Multipoles H atoms – monopoles, one bond directed dipole (D0) (up to quadrupoles) Second period elements (Li – F) – up to octupoles Third period elements (Na – Cl) – possibly up to hexadecapoles Heavier elements – up to hexadecapoles

  31. Jyväskylä Summer School on Charge Density August 2007 Refinement strategy for Multipoles Start with a simple model and gradually increase the complexity (flexibility). At each stage, check that any increase in flexibility results in a significant improvement. Apply full chemical and symmetry restraints (often the symmetry will be/ needs be only approximate). Think carefully about the local coordinate system which must be defined for all atoms. C atom m O atom m N atom mm2 H atom cyl Formamide HC(=O)NH2

  32. Jyväskylä Summer School on Charge Density August 2007 Refinement strategy for Multipoles Need to consult Table in the XD manual, which gives site symmetry restrictions on multipoles. • Tells us that • the local z-axis must be defined so it is perpendicular to the mirror plane • the allowed multipoles are (0,0), (1,1±), (2,2±), (2,0), (3,3±), (3,1±), (4,4±), (4,2±), (4,0) • The allowed multipoles are merely those which are symmetric w.r.t. the symmetry elements. • Sometimes crystallographic site symmetry mandates the use of these restrictions.

  33. Jyväskylä Summer School on Charge Density August 2007 Refinement strategy for Multipoles ATOM table ATOM ATOM0 AX1 ATOM1 ATOM2 AX2 R/L TP TBL KAP LMX SITESYM CHEMCON O(1) C(1) X O(1) N(1) Y R 2 1 1 4 m N(1) C(1) Z N(1) O(1) Y R 2 2 2 4 mm2 C(1) N(1) X C(1) O(1) Y R 2 3 3 4 m H(1) N(1) Z H(1) C(1) Y R 1 4 4 1 cyl H(2) N(1) Z H(2) C(1) Y R 1 4 4 1 cyl H(3) C(1) Z H(3) O(1) Y R 1 4 4 1 cyl KEY table RESET BOND N(1) H(1) 1.0 etc KEY xyz --U2-- ----U3---- ------U4------- M- -D- --Q-- ---O--- ----H---- O(1) 111 111111 0000000000 000000000000000 10 110 10011 0110011 100110011 N(1) 111 111111 0000000000 000000000000000 10 100 10010 0100010 100100010 C(1) 111 111111 0000000000 000000000000000 10 110 10011 0110011 100110011 H(1) 000 000000 0000000000 000000000000000 10 001 10000 0000000 000000000 H(2) 000 000000 0000000000 000000000000000 10 001 10000 0000000 000000000 H(3) 000 000000 0000000000 000000000000000 10 001 10000 0000000 000000000 . . . SCALE 1 END KEY

  34. Jyväskylä Summer School on Charge Density August 2007 Verification of refinement strategy The refined parameters need to be checked to see if they represent a physically sensible density. This can be done through (a) Low residual indices, R values and GOF. This is a necessary but not sufficient condition – many deficiencies in the model and data do not manifest in high residual indices. (b) Difference Fourier maps. This is an essential test – an ideal map is featureless. Deficiencies in the model often manifest in spurious features. (c) Anisotropic thermal parameters. The rigid bond test proposed by Hirshfeld should be checked at each stage. Typically we wish to see all dmsa < 0.001 Å2 for the covalently bonded pairs of atoms (except H atoms) – the DMSDA command in XDLSM. Differences of Mean-Squares Displacement Amplitudes (DMSDA) (1.E4 A**2) along interatomic vectors (*bonds) ATOM--> ATOM / DIST DMSDA ATOM / DIST DMSDA ATOM / DIST DMSDA O(1) C(1) * 1.2405 1 N(1) C(1) * 1.3193 -4 F. L. Hirshfeld (1976). Acta Cryst, A32, 239

  35. Jyväskylä Summer School on Charge Density August 2007 Deficiences of the Multipole Model The deficiencies of the multipole model have been much discussed in recent years. Mostly shows up as discrepancies in topological parameters when comparing experimental and theoretically derived densities. One well known case concerns polar covalent bonds. Possible reasons for discrepancies include (a) inadequate basis sets in theoretical studies (b) neglect of electron correlation (c) neglect of crystal environment (calculations mostly in gas phase) (d) deficiencies in multipole model, particularly the radial functions. Quantum calculations are usually undertaken using Gaussian basis sets. Coppens has noted that discrepancy between theory and experiment is less when Slater bases are used in theoretical calculations (ADF). The KRMM was one proposed way of reducing the influence of kappa refinement. C. Gatti, R. Bianchi, R. Destro & F. Merati (1992) J. Mol. Struct255 409. (alanine) A. Volkov, Y. Abramov, P. Coppens & C. Gatti (2000) Acta Cryst. A56, 332. (p-nitro-aniline) D. Stalke et al (2004) J. Phys. Chem. A108 9442. (S-N bonds)

  36. Jyväskylä Summer School on Charge Density August 2007 Deficiences of the Multipole Model “for chemically bound atoms, theoretically derived RDF’s are superior to those obtained from calculations on isolated atoms, even if differences ... do not manifest themselves in the usual figures of merit” B. Dittrich, T. Koritsanszky, A. Volkov, S. Mebs, P. Luger (2007) Angew Chemie. 46, 2935 T. Koritsanszky, A. Volkov (2004) Chem Phys Lett. 385, 431

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